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31

The Black-Scholes option pricing model provides a closed-form pricing formula $BS(\sigma)$ for a European-exercise option with price $P$. There is no closed-form inverse for it, but because it has a closed-form vega (volatility derivative) $\nu(\sigma)$, and the derivative is nonnegative, we can use the Newton-Raphson formula with confidence. Essentially, ...

28

I have worked on this topic extensively (pricing and calculating IV in production) and believe can offer an informed opinion. First of all Mathworks - the company that creates Matlab is not a trading firm so you should probably not rely on their advice so much. There are few closed form options pricing models, and all have practical shortcomings. Barone-...

28

You can directly imply a probability distribution from a volatility skew. Note that, for any terminal probability distribution $p(S)$ at tenor $T$, we have the model-free formula for the call price $C(K)$ as a function of strike $K$ $$C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS$$ Therefore we can write e^{rT} \...

25

Let $t_0, t_1, \ldots, t_n$ be observation dates, where $0=t_0 < \cdots < t_n = T$, and $\{S_t \mid t \geq 0\}$ be the equity price process without dividend payments. Then the realized variance is defined by \begin{align*} \frac{252}{n}\sum_{i=1}^n \ln^2 \frac{S_{t_i}}{S_{t_{i-1}}}. \end{align*} Note that, for sufficiently small $x$, \begin{align*} \...

22

Brenner and Subrahmanyam (1988) provided a closed form estimate of IV, you can use it as the initial estimate: $$\sigma \approx \sqrt{\cfrac{2\pi}{T}} . \cfrac{C}{S}$$

18

Consider a more financially plausible model than Black-Scholes: one where the stock can suddenly go bankrupt due to fraud, and the volatility varies over time. Neither model is perfect, but the new one (call it SVJ) will be "less wrong". Mathematically, we no longer have the Black-Scholes SDE based on a single stochastic generator $W$ $$\frac{dS}{S} = \... 17 Taking away all frictions and incomplentess of the market, the theory says that European Call and Puts do have the same implied volatility unless there is an arbitrage opportunity by put call parity$$ C(t,K) - P(t,K) = DF_t(F_t - K)\ . $$If you plug the Black-Scholes formula here for the prices of the call and the put, you will see that the equality only ... 16 It is a very simple procedure and yes, Newton-Raphson is used because it converges sufficiently quickly: You need to obviously supply an option pricing model such as BS. Plug in an initial guess for implied volatility -> calculate the the option price as a function of your initial iVol guess -> apply NR -> minimize the error term until it is sufficiently ... 16 You may want to first broadly categorize volatility models before comparing between them within each class, it does not make sense to compare standard deviation models with an implied vol model. I would broadly classify as follows: Historical realized volatility: Those include standard deviation (sum of squared deviations), realized range volatility ... 16 There is no "plain Black Scholes implied surface" because implied volatilities come from options market prices (calls and put). If you had a whole continuum of call prices C : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+, (T,K) \mapsto C(T,K) you would get a implied volatility function \sigma_I : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+ ... 15 Setting aside, that it's not pure riskless arbitrage, but rather statistical arbitrage: You can extract the profit by performing continuous delta hedging. If you constantly adjust your hedge position you gain/lose money by delta hedging. Being long option (gamma long), you sell at higher prices and buy at lower ones. Over the course of time you realize ... 15 I generally agree with @dm63's answer: A convex (concave) smile around the forward usually indicates and leptokurtic (platykurtic) implied risk-neutral probability density. Both situations can or cannot admit arbitrage. I provide you with two counterexamples to your statements. A volatility smile that is concave around the forward does not necessarily ... 13 It seems that you are thinking of the volatility as some sort of standard deviation of your stock price. It is not. In the BS model, \sigma\sqrt{T} is the standard deviation of the log-return \log(\frac{S_T}{S_0}). There is no mathematical upper bound to its standard deviation. There is also no mathematical problem with returns being negative either. ... 12 Yes it is a better way. Just take a look to figure 3, from Buss and Vilkov (2012, RFS): 11 The way market makers mark their volatility curves is by using models which 'fill in the gaps', i.e. they will make a price for a given option even if they do not believe this option is going to get a lot of volume. They are still willing to go long/short because they have a strategy to hedge their overall position (i.e. by managing their greeks and expiries)... 11 Implied volatility has very little to do with any particular pricing model, especially not much with BS. BS is a translation tool between prices and volatility, with its own many model deficiencies. I won't get into such model assumptions because my point is an entirely different one. Even the smile/smirk is entirely unrelated to the Black-Scholes model and ... 11 Along with Gatheral's book, I'd recommend reading Lorenzo Bergomi's "Stochastic Volatility Modelling". The first 2 chapters are available for download on his website. That being said, let me try to give you the basic picture. Below we assume that the equity forward curve F(0,t)=\Bbb{E}_0^\Bbb{Q}[S_t] is given for all t smaller than some relevant ... 10 My try to answer this question with some other questions: Is the BS model right? No. Is it useful: yes. Taking a traded price and the BS Model there is only one input factor that is not given by the market: the implied volatility. It is a measure to compare options across time and strike. Are there better models? yes. Those that you mention: The local vol ... 10 Upon close reading, this appears to be 3 (interesting) questions, not one. I'm not sure if the mods have the tools needed to split it up, so I'm just going to write down the three questions as I see them and then deal with them one by one. Note, it is simpler for me to talk about variance instead of volatility. This has no material impact on the answer. ... 10 Note that total implied variance defined as$$ V(T,K) = T\Sigma(T,K)^2 $$should be an increasing function of T. Otherwise you have a calendar arbitrage (sell the call with shorter expiry and buy the cheap longer one). If you interpolate linearly your implied volatility is$$ \Sigma(T,K) = w\Sigma(T_i,K) + (1-w)\Sigma(T_{i+1},K) $$with weight w = \... 10 What they gave you is Newton's formula. If you have a function f(x) then you can find the value x_0 such that f(x_0) = 0 by this method. It uses the derivative f' which in your case is the vega. Your function is:$$ f(x) = BS(x) - M $$where BS is the theoretical price with volatility x and M is the marketprice. Then f'(x) is the ... 9 VIX is a measure of implied volatility, specifically, model-free implied volatility, a concept originally developed by Demeterfi et. al. at Goldman Sachs in the 1990s. One of my recent questions, How to extrapolate implied volatility for out of the money options?, addressed some aspects of MFIV, and the papers mentioned in the question and answers will give ... 9 At strikes distant from the forward value, pretending that options have some meaningful implied volatility gets kind of silly. Options really have prices (both bids and offers), and we all just translate that to volatility because doing so provides a convenient normalization. Just to take one example, discrete price quoting completely obfuscates the ... 9 Scott Mixon argues in What Does Implied Volatility Skew Measure that among all measures of implied volatility skew, the (25 delta put volatility - 25 delta call volatility)/50 delta volatility is the most descriptive and least redundant (volatility is Black-Scholes implied volatility). His paper, recently published in the Journal of Derivatives, gives a ... 9 A very popular choice for mean reversion is the Ornstein–Uhlenbeck process (here in discretized form):$$L_{t+1}-L_t=\alpha(L^*-L_t)+\sigma\epsilon_t Here you see that the level change is governed by some parameter $\alpha$, the mean reversion rate (or speed), and the distance between the long run mean $L^*$ and the actual level $L_t$ plus some noise. A ...

9

The idea of regime switching in volatility is rooted in the observation that volatility is usually fairly consistent and "mild", and occasionally very high, say during a market crash. The concept goes further, though. Not only does the volatility level differ markedly in different regimes, but the behavior of volatility does as well (degree of mean reversion,...

9

For an option with price $C$, the P$\&$L, with respect to changes of the underlying asset price $S$ and volatility $\sigma$, is given by \begin{align*} P\&L = \delta \Delta S + \frac{1}{2}\gamma (\Delta S)^2 + \nu \Delta \sigma, \end{align*} where $\delta$, $\gamma$, and $\nu$ are respectively the delta, gamma, and vega hedge ratios. Then it is clear ...

9

From an equities perspective, there are two concepts that should not be confused in my opinion and context should make the distinction self-explicit: Forward variance swap volatility (A) Forward implied volatility smile (B) I really recommend reading Bergomi's "Stochastic Volatility Modeling" which is an excellent book for equity practitioners. The topics ...

8

OptionMetrics uses a kernel smoothing algorithm to interpolate the volatility surface. Their assumptions tend to be based on the academic consensus and have become somewhat industry standard, so the real answer to your question may be that there really is no good functional form.

8

Well as far as I know it is a really hard but interesting question. Asymptotics of smile in the strike direction is not known in a model free way as far as I know. I think I can remember that nevertheless you have upper and lower bounds if you know something about the underlying dynamics and especially the first moment of explosion. I can't remember the ...

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